Time integration

Time-stepping methods are algorithms for the numerical solution of ordinary differential equations and the time evolution of PDE discretizations. I work on advanced time-integration techniques including general linear methods, implicit–explicit (IMEX) schemes, and multirate time-stepping algorithms. Many of these methods are implemented in PETSc and DESolve.

Software and project links for this research area.

DESolve website DESolve GitHub PETSc Time integration project

Selected journal publications

Johann Rud*, Max Heldman, Emil M. Constantinescu, Qi Tang, and Xian-Zhu Tang. Scalable implicit solvers with dynamic mesh adaptation for a relativistic drift-kinetic Fokker-Planck-Boltzmann model. Journal of Computational Physics, Vol. 507; Pages 112954, 2024. [DOI] [arXiv] [PDF]
Shinhoo Kang, Alp Dener, Aidan Hamilton, Hong Zhang, Emil M Constantinescu, and Robert Jacob. Multirate partitioned Runge-Kutta methods for coupled Navier-Stokes equations. Computers & Fluids, Vol. 264(15); Pages 105964, 2023. [DOI] [arXiv] [PDF]
Shinhoo Kang and Emil M Constantinescu. Learning subgrid-scale models with neural ordinary differential equations. Computers and Fluids, In Press, Vol. 261; Pages 105919, 2023. [DOI] [arXiv]
Daniel Adrian Maldonado, Emil M Constantinescu, Junbo Zhao, and Mihai Anitescu. Computationally efficient power system maximum transient linear growth estimation. Submitted, 2023. [arXiv] [PDF]
Hong Zhang, Emil M Constantinescu, and Barry F. Smith. PETSc TSAdjoint: A discrete adjoint ODE solver for first-order and second-order sensitivity analysis. SIAM Journal of Scientific Computing, Vol. 44; Pages C1-C24, 2022. [DOI] [arXiv] [PDF]
Youngdae Kim, Debojyoti Ghosh, Emil M Constantinescu, and Ramesh Balakrishnan. GPU-accelerated WENO schemes for the DNS of compressible turbulent flows. Computers & Fluids, Vol. 251; Pages 105744, 2022. [DOI] [arXiv]
Shinhoo Kang and Emil M Constantinescu. Entropy-preserving and entropy-stable relaxation IMEX and multirate time-stepping methods. Journal of Scientific Computing, Vol. 93(23), 2022. [DOI] [arXiv]
Luisa D'amore, Emil M Constantinescu, and Luisa Carracciuolo. A scalable space-time domain decomposition approach for solving large scale non linear regularized inverse ill posed problems in 4D variational data assimilation,. Springer Journal of Scientific Computing, 2022. [DOI]
Hong Zhang, Zhengyu Liu, Emil M Constantinescu, and Robert Jacob. Stability analysis of coupled advection-diffusion models with bulk interface condition. Journal of Scientific Computing, Vol. 93(33), 2022. [DOI]
Hong Zhang and Emil M Constantinescu. Optimal checkpointing for adjoint multistage time-stepping schemes. Journal of Computational Science, Vol. 366, 2022. [DOI] [arXiv]
Alina Kononov, Cheng-Wei Lee, Tatiane Pereira dos Santos, Brian Robinson, Yifan Yao, Yi Yao, Xavier Andrade, Andrew David Baczewski, Emil M Constantinescu, Alfredo Correa, Yosuke Kanai, Norman Modine, and Andre Schleife. Electron dynamics in extended systems within real-time time-dependent density functional theory,. MRS Communications, Pages 1-13, 2022. [DOI] [arXiv] [PDF]
Adrian Maldonado, Emil M Constantinescu, Hong Zhang, Vishwas Rao, and Mihai Anitescu. Trust-region approximation of extreme trajectories in power system dynamics. IEEE Transactions on Power Systems, Vol. 37(5); Pages 3937-3946, 2022. [DOI] [arXiv]
Shinhoo Kang, Emil M Constantinescu, Hong Zhang, and Robert Jacob. Mass-conserving implicit-explicit methods for coupled compressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering (CMAME), Vol. 384; Pages 113988, 2021. [DOI] [arXiv]
Emil M Constantinescu. Implicit extensions of an explicit multirate Runge–Kutta scheme. Applied Mathematics Letters, Vol. 128; Pages 107871, 2021. [DOI] [arXiv]
Hong Zhang, Zhengyu Liu, Emil M Constantinescu, and Robert Jacob. Stability analysis of interface conditions for ocean-atmosphere coupling. Springer Journal of Scientific Computing, Vol. 84(44), 2020. [DOI] [arXiv] [PDF]
Daniel S. Abdi, Francis X. Giraldo, Emil M. Constantinescu, Lester E. Carr III, Lucas C. Wilcox, and Timothy Warburton. Acceleration of an implicit-explicit non-hydrostatic unified model of the atmospheric (NUMA) on manycore processors. International Journal of High Performance Computing Applications, Vol. 33(2); Pages 242-267, 2019. [DOI] [arXiv] [PDF]
Valeria Mele, Emil M Constantinescu, Luisa Carracciuolo, and Luisa D'Amore. A PETSc parallel-in-time solver based on MGRIT algorithm. {"Concurrency and Computation"=>"Practice and Experience"}, Vol. 30(24); Pages e4928, 2018. [DOI]
Simone Marras, Michal A. Kopera, Emil M Constantinescu, Jenny Suckale, and Francis X. Giraldo. A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations. Advances in Water Resources, Vol. 114; Pages 45-63, 2018. [DOI] [arXiv]
Shrirang Abhyankar, Jed Brown, Emil M Constantinescu, Debojyoti Ghosh, Barry F. Smith and Hong Zhang. PETSc/TS: A modern scalable ODE/DAE solver library. 2018. [arXiv]
Emil M Constantinescu. Generalizing global error estimation for ordinary differential equations by using coupled time-stepping methods. Journal of Computational and Applied Mathematics, Vol. 332(C); Pages 140-158, 2018. [DOI] [arXiv]
Shrirang Abhyankar, Emil M Constantinescu, Barry Smith, Alexander J. Flueck, and Daniel A. Maldonado. Parallel dynamics simulation using a Krylov-Schwarz linear solution scheme. IEEE Transactions on Smart Grid Special Issue on High Performance Computing (HPC) Applications for a More Resilient and Efficient Power Grid, Vol. 8(3); Pages 1378-1386, 2017. [DOI]
Noemi Petra, Cosmin G. Petra, Zheng Zhang, Emil M Constantinescu, and Mihai Anitescu. A Bayesian approach for parameter estimation with uncertainty for dynamic power systems. IEEE Transactions on Power Systems, Vol. 32(4); Pages 2735-2743, 2017. [DOI] [arXiv]
Hong Zhang, Shrirang S. Abhyankar, Emil M Constantinescu, and Mihai Anitescu. Discrete adjoint sensitivity analysis of power system dynamics. {"IEEE Transactions on Circuits and Systems--I"=>"Regular Papers"}, Vol. 64(5); Pages 1247-1259, 2017. [DOI]
Debojyoti Ghosh and Emil M Constantinescu. A well-balanced conservative finite-difference algorithm for atmospheric flows. AAIA Journal, Vol. 54(4); Pages 1370-1385, 2016. [DOI] [PDF]
Debojyoti Ghosh and Emil M Constantinescu. Semi-implicit time integration of atmospheric flows with characteristic-based flux partitioning. SIAM Journal on Scientific Computing (SISC), Vol. 38(3); Pages A1848-A1875, 2016. [DOI] [arXiv] [PDF]
Debojyoti Ghosh, Emil M Constantinescu, and Jed Brown. Efficient implementation of non-linear compact schemes on massively-parallel platforms. SIAM Journal on Scientific Computing (SISC), Vol. 37(3); Pages C354-C383, 2015. [DOI]
Francis X Giraldo, James F. Kelly, and Emil M Constantinescu. Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM Journal on Scientific Computing (SISC), Vol. 35(5); Pages B1162-B1194, 2013. [DOI]
Emil M Constantinescu and Adrian Sandu. Extrapolated multirate methods for differential equations with multiple time scales. Journal of Scientific Computing, Vol. 56(1); Pages 28-44, 2013. [DOI]
David E. Keyes, Lois Curfman McInnes, Carol Woodward, William Gropp, Eric Myra, Michael Pernice, John Bell, Jed Brown, Alain Clo, Jeffrey Connors, Emil M Constantinescu, Don Estep, Kate Evans, Charbel Farhat, Ammar Hakim, Glenn Hammond, Glen Hansen, Judith Hill, Tobin Isaac, Xiaomin Jiao, Kirk Jordan, Dinesh Kaushik, Efthimios Kaxiras, Alice Koniges, Kihwan Lee, Aaron Lott, Qiming Lu, John Magerlein, Reed Maxwell, Michael McCourt, Miriam Mehl, Roger Pawlowski, Amanda Peters Randles, Daniel Reynolds, Beatrice Rivière, Ulrich Rüde, Tim Scheibe, John Shadid, Brendan Sheehan, Mark Shephard, Andrew Siegel, Barry Smith, Xianzhu Tang, Cian Wilson, and Barbara Wohlmuth. Multiphysics simulations: Challenges and opportunities. International Journal of High Performance Computing Applications, Vol. 27(1); Pages 4-83, 2013. [DOI]
Emil M Constantinescu and Adrian Sandu. Optimal strong-stability-preserving general linear methods. SIAM Journal on Scientific Computing (SISC), Vol. 32(5); Pages 3130-3150, 2010. [DOI]
Emil M Constantinescu and Adrian Sandu. Extrapolated implicit-explicit time stepping. SIAM Journal on Scientific Computing (SISC), Vol. 31(6); Pages 4452-4477, 2010. [DOI]
Emil M Constantinescu. On the order of general linear methods. Applied Mathematics Letters, Vol. 22(9); Pages 1425-1428, 2009. [DOI]
Adrian Sandu and Emil M Constantinescu. Multirate explicit Adams methods for time integration of conservation laws. Journal of Scientific Computing, Vol. 38(2); Pages 229-249, 2009. [DOI] [PDF]
Emil M Constantinescu, Adrian Sandu, and Gregory R. Carmichael. Modeling atmospheric chemistry and transport with dynamic adaptive resolution. Computational Geosciences, Vol. 12(2); Pages 133-151, 2008. [DOI]
Emil M Constantinescu and Adrian Sandu. Multirate timestepping methods for hyperbolic conservation laws. Journal of Scientific Computing, Vol. 33(3); Pages 239-278, 2007. [DOI] [PDF]

Proceedings / presentations

Sheng Lei, Adrian Maldonado, Emil M Constantinescu, Junbo Zhao, Someya Yarahmadi, Lamine Mili, and Mihai Anitescu. A novel continuum approximation to power system electromechanical dynamics. IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT), Pages 1-5, 2022. [DOI]
Adrian Maldonado and Emil M Constantinescu. Analysis of the growth of small perturbations in power system dynamics. IEEE PSCC 2022, 2022.
Hong Zhang and Emil M Constantinescu. Revolve-based adjoint checkpointing for multistage time integration. ICCS 2021 (International Conference on Computational Science 2021), Pages 451-464, 2021.
Valeria Mele, Diego Romano, Emil M Constantinescu, Luisa Carracciuolo, and Luisa D'Amore. Performance evaluation for a PETSc parallel-in-time solver based on the MGRIT algorithm. Euro-Par 2018; 24th International European Conference on Parallel and Distributed Computing, Turin, Italy, August 27-31, 2018, 2018.
Shrirang Abhyankar, Emil M Constantinescu, and Alexander Flueck. Variable-step multi-stage integration methods for fast and accurate power system dynamics simulation. Proceedings of IREP 2017 Symposium on X Bulk Power Systems Dynamics and Control Symposium, Espinho, August 27 - September 1, 2017, 2017.
Pierre-Louis Guhur, Emil M Constantinescu, Debojyoti Ghosh, Tom Peterka, and Franck Cappello. Detection of silent data corruption in adaptive numerical integration solvers. IEEE Cluster 2017, September 5-8, 2017, Hawaii, USA, Pages 592-602, 2017.
Pierre-Louis Guhur, Hong Zhang, Tom Peterka, Emil M Constantinescu, and Franck Cappello. Lightweight and accurate silent data corruption detection in ordinary differential equation solvers. Euro-Par 2016; 22nd International European Conference on Parallel and Distributed Computing, Grenoble, France, August 22-26, 2016; Proceedings: Springer Euro-Par 2016 Parallel Processing, Theoretical Computer Science and General Issues, Pages 644-656, 2016. [DOI]
Debojyoti Ghosh and Emil M Constantinescu. Nonlinear compact finite-difference schemes with semi-implicit time stepping. Springer's Lecture Notes in Computational Science and Engineering (LNCSE) Series, ICOSAHOM 2014, Vol. 106; Pages 237-245, 2015. [DOI]
Debojyoti Ghosh and Emil M Constantinescu. Well-balanced formulation of gravitational source terms for conservative finite-difference atmospheric flow solvers. 7th AIAA Atmospheric and Space Environments Conference (Aviation Forum), June 22-26, 2015, Dallas, TX, 2015.
Shrirang Abhyankar, Barry Smith, and Emil M Constantinescu. Evaluation of overlapping restricted additive Schwarz preconditioning for parallel solution of very large power flow problems. Proceedings of the 3rd International Workshop on High Performance ComputingNetworking and Analytics for the Power Grid, Pages 5:1-5:8, 2013. [DOI]
Barry Smith, Lois Curfman McInnes, Emil M Constantinescu, Mark Adams, Satish Balay, Jed Brown, Matthew Knepley, and Hong Zhang. PETSc's software strategy for the design space of composable extreme-scale solvers. DOE Exascale Research ConferencePortlandORApril 16-18, 2012.
Emil M Constantinescu and Adrian Sandu. On multirate numerical integration methods. (invited) The European Consortium for Mathematics in Industry (ECMI-2008) in Progress in Industrial Mathematics at ECMI 2008, University College London, England, June 30 - July 4, 2008, Vol. 15(2); Pages 341-347, 2010. [DOI]
Emil M Constantinescu and Adrian Sandu. Explicit time stepping methods with high stage order and monotonicity properties. Lecture Notes in Computer Science, International Conference on Computational Science (ICCS), Baton Rouge, Louisiana, May 25-27, 2009, 2009.
Adrian Sandu and Emil M Constantinescu. Multirate time discretizations for hyperbolic partial differential equations. International Conference of Numerical Analysis and Applied Mathematics 2009 (ICNAAM 2009), Crete, Greece, September 18-22, 2009, Vol. 1168(1); Pages 1411-1414, 2009. [DOI]
Emil M Constantinescu, Wenyuan Liao, and Adrian Sandu. Mesh refinement strategies in air quality modeling. High Performance Computing Symposium, San Diego, CA, April 2005, Pages 158-163, 2005.
Emil M Constantinescu and Adrian Sandu. On adaptive mesh refinement for atmospheric pollution models. International Conference on Computational Science (ICCS), Atlanta, GA, May 22-25, 2005, Springer-Verlag in Lecture Notes in Computer Science, Pages 798-805, 2005.

Technical reports

Satish Balay, S. Abhyankar, Mark F. Adams, Jed Brown, P. Brune, K. Buschelman, Emil M Constantinescu, Lisandro Dalcin, S. Benson, A. Dener, and others. PETSc/TAO users manual revision 3.24. Argonne National Laboratory (ANL), Argonne, IL (United States), 2025.
Debojyoti Ghosh, Emil M Constantinescu, and Jed Brown. Scalable nonlinear compact schemes. 2014. [PDF]
D.E. Keyes, L.C. McInnes, C. Woodward, W.D. Gropp, E. Myra, M. Pernice, J. Bell, J. Brown, A. Clo, J. Connors, E. Constantinescu, D. Estep, K. Evans, C. Farhat, A. Hakim, G. Hammond, G. Hansen, J. Hill, T. Isaac, X. Jiao, K. Jordan, D. Kaushik, E. Kaxiras, A. Koniges, K. Lee, A. Lott, Q. Lu, J. Magerlein, R. Maxwell, M. McCourt, M. Mehl, R. Pawlowski, A. Peters, D. Reynolds, B. Rivière, U. Rüde, T. Scheibe, J. Shadid, B. Sheehan, M. Shephard, A. Siegel, B. Smith, X. Tang, C. Wilson, B. Wohlmuth. Multiphysics simulations: challenges and opportunities. 2012.
Emil M Constantinescu and Adrian Sandu. Optimal explicit strong-stability-preserving general linear methods: Complete results. 2009.
Emil M Constantinescu and Adrian Sandu. Achieving very high order for implicit explicit time stepping: extrapolation methods. 2009.
Emil M Constantinescu and Adrian Sandu. On extrapolated multirate methods. 2008.
Emil M Constantinescu and Adrian Sandu. Update on multirate timestepping methods for hyperbolic conservation laws. 2007. [PDF]
Adrian Sandu and Emil M Constantinescu. Multirate explicit Adams methods for time integration of conservation laws. 2007. [PDF]
Emil M Constantinescu and Adrian Sandu. Multirate timestepping methods for hyperbolic conservation laws. 2006. [PDF]